Percolation analysis - Revision history

Rziff at 18:12, 15 October 2023

2023-10-15T18:12:02Z

Carl McBride: Added a book

2011-05-30T13:00:13Z

Added a book

← Older revision		Revision as of 15:00, 30 May 2011
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	==References==		==References==
	<references/>		<references/>
			;Related reading
			*Dietrich Stauffer and Ammon Aharony "Introduction To Percolation Theory", 2nd Edition, CRC Press (1994) ISBN 9780748402533
	[[Category: Confined systems]]		[[Category: Confined systems]]

Nice and Tidy: /* Percolation threshold and critical thermodynamic transitions */ Added and deleted white-space.

2009-11-19T14:11:04Z

Percolation threshold and critical thermodynamic transitions: Added and deleted white-space.

← Older revision		Revision as of 16:11, 19 November 2009
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	In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>		In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
	<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>		<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories", Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
	<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices", Physical Review B '''40''', 5007 - 5014 (1989) ] </ref>		<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices", Physical Review B '''40''', 5007-5014 (1989)] </ref>
	. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).		. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).

Noe: /* Percolation threshold and critical thermodynamic transitions */ minor edit

2009-11-18T11:30:08Z

Percolation threshold and critical thermodynamic transitions: minor edit

← Older revision		Revision as of 13:30, 18 November 2009
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	In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>		In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
	<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>		<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
	<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" Physical Review B '''40''', 5007 - 5014 (1989) </ref>		<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices", Physical Review B '''40''', 5007 - 5014 (1989) ] </ref>
	. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).		. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).

Noe: /* Percolation threshold and critical thermodynamic transitions */ new reference added

2009-11-18T11:28:59Z

Percolation threshold and critical thermodynamic transitions: new reference added

← Older revision		Revision as of 13:28, 18 November 2009
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	In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>		In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
	<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>		<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
	<ref name=hu >		<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" Physical Review B '''40''', 5007 - 5014 (1989) </ref>
	[http://dx.doi.org//10.1103/PhysRevB.40.5007
	Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" ~~Phys. Rev.~~ B 40, 5007 - 5014 (1989) </ref>
	. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).		. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).

Noe: /* Percolation threshold and critical thermodynamic transitions */

2009-11-18T11:27:31Z

Percolation threshold and critical thermodynamic transitions

← Older revision		Revision as of 13:27, 18 November 2009
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	In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>		In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
	<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).		<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
			<ref name=hu >
			[http://dx.doi.org//10.1103/PhysRevB.40.5007
			Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" Phys. Rev. B 40, 5007 - 5014 (1989) </ref>
			. In these case [[cluster algorithms\|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms\|cluster algorithms]] page for more details).

	==References==		==References==
	<references/>		<references/>
	[[Category: Confined systems]]		[[Category: Confined systems]]

Nice and Tidy: Slight tidy of references.

2009-10-05T10:00:55Z

Slight tidy of references.

← Older revision		Revision as of 12:00, 5 October 2009
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	Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.		Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
	Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009) ~~[9 pages]~~] </ref>		Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009)]</ref>

	== Connectivity rules ==		== Connectivity rules ==
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	A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.		A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
	These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>		These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
	<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) ~~[16 pages]~~ ] </ref> for details).		<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).

	=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===		=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

Noe: /* Lattice and continuum (off-lattice) models */

2009-10-02T13:06:28Z

Lattice and continuum (off-lattice) models

← Older revision		Revision as of 15:06, 2 October 2009
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	Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.		Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
	Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", ~~Phys. Rev.~~ E 80, 041101 (2009) [9 pages]] </ref>		Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009) [9 pages]] </ref>

	== Connectivity rules ==		== Connectivity rules ==

Noe: /* Lattice and continuum (off-lattice) models */ new ref

2009-10-02T13:05:27Z

Lattice and continuum (off-lattice) models: new ref

← Older revision		Revision as of 15:05, 2 October 2009
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	Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.		Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
	Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref>		Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Phys. Rev. E 80, 041101 (2009) [9 pages]] </ref>

	== Connectivity rules ==		== Connectivity rules ==

Noe: /* Lattice and continuum (off-lattice) models */

2009-10-01T17:18:26Z

Lattice and continuum (off-lattice) models

← Older revision		Revision as of 19:18, 1 October 2009
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	Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.		Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
	Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more realistic to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref>		Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref>

	== Connectivity rules ==		== Connectivity rules ==

← Older revision		Revision as of 20:12, 15 October 2023
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	This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>		This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato">Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

	==Sites, bonds, and clusters ==		==Sites, bonds, and clusters ==
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	== Percolation and finite-size scaling analysis ==		== Percolation and finite-size scaling analysis ==
	=== Example: Site-percolation on a square lattice ===		=== Example: Site-percolation on a square lattice ===
	Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>		Let us consider a standard example of percolation theory, <ref name=deng> [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
	a two-dimensional [[building up a square lattice\|square lattice]] in which:		a two-dimensional [[building up a square lattice\|square lattice]] in which:
	* Each site of the lattice can be occupied (by one ''particle'') or empty.		* Each site of the lattice can be occupied (by one ''particle'') or empty.
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	== Computation of the percolation threshold ==		== Computation of the percolation threshold ==
	A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.		A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
	These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng~~' > <~~/~~ref~~> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>		These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name=deng/> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
	<ref name=~~Newmann~~> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).		<ref name=Newman> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).

	=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===		=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===
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	:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>		:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

	where <math> b </math> is a [[Critical exponents \|critical exponent]] (See Refs. <ref name="Stauffer"~~> <~~/~~ref~~> <ref name="Torquato~~"> <~~/~~ref~~> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is		where <math> b </math> is a [[Critical exponents \|critical exponent]] (See Refs. <ref name="Stauffer"/> <ref name=Torquato/>for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
	possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.		possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.